The cochlear ear horn: geometric origin of tonotopic variations in auditory signal processing

While separating sounds into frequency components and subsequently converting them into patterns of neural firing, the mammalian cochlea processes signal components in ways that depend strongly on frequency. Indeed, both the temporal structure of the response to transient stimuli and the sharpness of frequency tuning differ dramatically between the apical and basal (i.e., the low- and high-frequency) regions of the cochlea. Although the mechanisms that give rise to these pronounced differences remain incompletely understood, they are generally attributed to tonotopic variations in the constituent hair cells or cytoarchitecture of the organ of Corti. As counterpoint to this view, we present a general acoustic treatment of the horn-like geometry of the cochlea, accompanied by a simple 3-D model to elucidate the theoretical predictions. We show that the main apical/basal functional differences can be accounted for by the known spatial gradients of cochlear dimensions, without the need to invoke mechanical specializations of the sensory tissue. Furthermore, our analysis demonstrates that through its functional resemblance to an ear horn (aka ear trumpet), the geometry of the cochlear duct manifests tapering symmetry, a felicitous design principle that may have evolved not only to aid the analysis of natural sounds but to enhance the sensitivity of hearing.


Hydrodynamics of the 3-D cochlea
For simplicity, we assume that the cross-section of the scalae is a rectangle of width w and height h (Fig. S1). Nevertheless, the approximations, derivations, and results shown here also apply to models where the shape of the cochlear crosssection is different (e.g., circular). We consider a wave elicited by a tone of angular frequency ω; to simplify the notation, the dependence on ω is generally omitted. The flexible part of the cochlear partition (CP) spans a fraction of the cochlea width, and we make no assumption about whether the CP is centered in the radial direction.
Laplace's equation for the pressure in the fluid is Employing a separation of variables, the Laplace equation yields where κ x,y,z indicates the local wavenumber in the corresponding direction. The above equation states that the pressure at any point in the cochlea can be determined as a superposition of * altoe@usc.edu † also at Department of Physics & Astronomy, University of Southern California, Los Angeles, CA; christopher.shera@usc.edu Given κ x , the amplitude of each mode depends on the profile of the transverse motion of the CP in the radial (y) direction. Neglecting compression waves traveling at the speed of sound in water the solution for the pressure is largely dominated by the mode defined by κ y = 0 because the width of the CP is on the order of 100 µm and the transverse displacement produced by low-level sounds is on the order of few nm. With these simplifications, one obtains In the limit that the space constant characterizing the tapering of the scalae height is larger than the height of the scalae, we can employ the averaging procedure detailed by Duifhuis [4] for 2D box models. Following Duifhuis and introducing the scalaeaveraged pressure,P (x), one can apply Newton's second law and mass conservation to obtain where the fluids are assumed inviscid, ρ is the fluid density, S(x) is the effective acoustic crosssectional area of the scalae, b(x) the width of the CP, V BM = Y CP P 0 , and P 0 = αP . Uncoupling these equations yields Equation (S.8) can be simplified by introducing the change of variables defined by (S.12) Converting this solution back into the x domain yields as given in the main text.

Wavefront delay and high-frequency cut-off
For what follows, it proves convenient to rewrite the equations in terms of frequency normalized to the local CF using the variable β(x, f ) = f /CF(x). Using the tonotopic map, this becomes In the base of the cochlea, where simple quantitative statements about the relationship between wavefront delay and cochlear dimensions can be made, the approximations η(x) ≈ 1 and dβ/dx ≈ β/ work well. In the apex, although the equations can be solved, the low frequency "bend" of the cochlear map complicates the solutions to the point that there is no advantage to showing those over purely numerical results.
To determine the wave-front delay and its relation to cut-off, it suffices to ignore the prefactor in Eq. (S.13) and focus on the complex exponential. Defining κ β as the wavenumber in the β domain, 16) enables one to rewrite the complex exponential (e iφ ) as At tail frequencies, where κ β is real, the group delay in periods of the CF is given by The tail-frequency approximation to the wavenumber discussed in the main text [Eq. (9)] implies that so that as given in the main text. Far enough from the helicotrema, the onset of the CP click response is dominated by components well below CF, and hence τ wf coincides with the onset latency of the response (i.e., with the wavefront delay of the traveling wave). As one approaches the helicotrema in the apex, both m(x)/k(x) and dx/dβ decrease, and therefore the wavefront delay also decreases. For an exponential frequency map, given the constraints on m and k deduced from the anatomical data, one obtains τ wf ∝ Ω(ω where Ω indicates asymptotic growth at least as fast as its argument. As frequency decreases, the low-frequency bend of the tonotopic map produces a steeper decline in the value of τ wf . When the wavenumber κ β has a significant negative imaginary part, the complex exponential e −i κ β dβ dominates the behavior of the high-frequency cut-off observed in mechanical and neural data. For simplicity, we assume that α ≈ 1 (i.e., the long-wave approximation); this approximation appears not unsatisfactory in the apex, where the scalae radius and the short-wave gain factor are small. We can then write the wavenumber in the form The cut-off occurs becauseŶ CP has a positive imaginary part above CF (i.e., the mechanics of the CP contain a significant dissipative component). The complex exponential in the pressure becomes indicating an exponential decay of pressure beyond the peak. Equation (S.22) implies that the larger the wave-front delay, τ wf , the more steeply the pressure decreases above CF; conversely, the smaller τ wf , the gentler the cut-off.

Model of the CP admittance
To a first approximation, we assume that the center of mass of the cochlear partition moves in proportion to the BM. We therefore employ the BM admittance, first deduced by Zweig [24], that we have previously shown provides an excellent fit to BM transfer functions measured in the base of the gerbil and in the apex of the mouse cochlea [1] . In particular, we assume that , (S.23) where τ is the strength of the active force, m a is the acoustic mass (or mass per unit area) of the BM (see [1]), and ζ is the damping factor. Rewriting this in terms of the scaling variable β = ω/ω c yields where the CP stiffness k = m a ω 2 c . Imposing the constraint that τ and ζ be constant ensures that the local micromechanics, as reflected in the ratio between BM velocity and driving pressure, retain the same sharpness of tuning along the cochlea. The parameters for cat are τ = 1.3 and ζ = 0.12. Consistent with our analysis, we assume that partition stiffness (k) and fluid acoustic mass (m) vary with position as deduced from the experimental data, with the apical-basal transition at CF a|b ≈ 3 kHz [20].

Numerical calculation of the short-wave gain factor
Having defined the geometry of the cochlea and the admittance of the partition, we can numerically compute the model's responses. Unfortunately, this is not as straight-forward as it appears, because the solution for P 0 (x) given by Eq. (5) is only formal. Note that P 0 (x) depends on α(x)-both directly via the prefactor and indirectly via the wavenumber, since κ depends on α [see Eq. (6)]-but that α(x), in turn, depends on P 0 (x) via the definition α = P 0 /P . Thus, the solution must be obtained by iteration [19].
To determine the value of α(x), we adopt the WKB approximation to α(x) given by Eq. (6) and solve by iteration the following pair of equations: At frequencies below the local CF, the iterative numerical solution obtained by starting from the long-wave approximation rapidly converges. Above CF, however, where Y CP is dominated by dissipative forces [14], solution is complicated by the existence of multiple solutions and the iteration can fail to converge. Physically, the multiple solutions represent evanescent modes (e.g., standing waves in the z direction) and are responsible for the sharp peaks and notches often observed in BM transfer functions at frequencies above CF (see [22,24]). In addition, the WKB approximation to α we employ [Eq. (6)] assumes a single mode and therefore breaks down well above CF (see [23]). We largely circumvent these problems in our calculations by forcing the iteration to converge on the value of κ whose real and imaginary parts have the same sign as the wavenumber obtained using the long-wave approximation, which neglects evanescent modes. Figure S2A,B show values of α(x) for waves of different frequencies as computed in the model used to obtain the transfer functions in Fig. 3B. Figure S2C,D show the corresponding real and imaginary parts of the wavenumber (κ). shows the value of 1/h(x) along the cochlea: when the wavenumber is smaller than this value, wavelengths are larger than the scala height, and the long-wave approximation is expected to hold well. When the wavenumber is larger than 1/h(x), the long-wave approximation breaks down and the pressure driving the motion of the CP undergoes an effective boost due to the short-wave hydrodynamics (see panel A and [24,1]). Note that in the long-wave region, the curves in panel C are approximately parallel, reflecting the approximate scaling behavior of the wavenumber at tail frequencies. Note also that at apical locations the pressure wave remains effectively long-wave, never entering a short-wave region. For visual clarity, the values of α and κ are shown only at locations where the magnitude of the BM transfer function is within 80 dB of its peak value.

Results in a gerbil model
Because of the wealth of information about apical-basal differences in the cat [e.g., 6,8,2,17], we presented the results of a model tailored to the this species in the main text. We show here that analogous results are obtained in a model tailored to the gerbil cochlea, where a comparison with mechanical data is possible. For the ger-bil, we assume a Greenwood-type tonotopic map with γ = 0.6 [5,9] and a total cochlear length of 12.5 mm. The model parameters were adjusted slightly from [1] (τ wf = 1.1, τ = 1.3 and ζ = 01.3) in order to yield a better match to measurements of the sharpness of tuning obtained from auditory-nerve fibers (ANFs) in the base of the cochlea. As a result, the model transfer function shown in Fig. S3A,B is somewhat more sharply tuned than the BM transfer function measured at 40 dB SPL. We are not too concerned about this discrepancy, both because the difference between model prediction and data (roughly 4 dB at the peak) is comparable to those observed between animals of the same species (see Fig. 3B of [1]) and because the experimental data were obtained at sound levels likely to be in the compressive regime [21], where tuning is broader than at sound levels near ANF threshold. Figure S3C shows that the model provides a good match to the sharpness of tuning measured in the auditory nerve [15]. For completeness, Figs. S3D,E show the magnitude and phase of a BM transfer function obtained near the apical end of the gerbil model (CF ≈ 1 kHz). The shape of the transfer function magnitude in Fig. S3D resembles those obtained in the apex of the cat model (see Fig.3B in main text and Fig. S5).

Active forces in the base and in the apex
The active force term in the model of the CP partition is This mathematical term provides only a phenomenological description of the effect of the active cochlear process on the response of the BM, but its mathematical form is firmly based on the experimental data [24,1]. We have previously shown that this active term provides a compelling phenomenological description of OHC forces, yielding nonlinear model responses that are strikingly similar to mechanical responses measured near the reticular lamina (top of the   OHCs) in the 9-10 kHz region of the mouse cochlea [1].
Although the local mechanics of the partition are described by the same equation throughout the cochlea, the manifestation of the active force in the model appears qualitatively different in the base and in the apex (Fig. S4). This difference arises because of apical-basal differences in the frequency tuning of the transpartition pressure resulting from the global geometry of the cochlea.
In both the apex and the base, the active force is more broadly tuned than the BM. At basal locations (Fig. S4A), the differences in tuning between the active force and the BM appear qualitatively similar to those observed between the reticular lamina (RL) and BM motion [11,7,3]. In the apex (Fig. S4B), the active force is much more broadly tuned and peaks at a frequency higher than the BM. For a visual comparison of tuning, the dashed line in Fig. S4B shows the frequency response of the active force on an axis normalized to its best frequency. That is, our model suggests that the broad tuning of the internal organ of Corti motion measured in the apex of the guinea pig [10]-and the apparent mismatch between neural and mechanical tuningresults from the broad tuning of the OHC forces rather than the broad tuning of the overall transverse motion of the partition. The truth of this model-based conjecture remains an interesting empirical question.

Appendix G
Effects of the active process on cochlear hydrodynamics As a traveling wave approaches its best place, the cochlear amplifier acts not only to increase its amplitude but also to decrease its group velocity, thereby increasing the group delay of the response [e.g., 13]. This coordinated action is a consequence of causality, as manifest through the Kramers-Krönig relations (see [16]). Physically, this means that the active process modifies both the real and imaginary parts of the wavenumber near the best place, effectively exploiting cochlear hydrodynamics to boost the the transpartition pressure wave via spatially distributed amplification [1]. Equation (S.25) implies that the wavenumber, κ, and the short-wave pressure gain factor, α, are intimately related. Both depend strongly on the radius of the scalae, h. As a result, our analysis reveals that the global, tapered geometry of the cochlea introduces a significant spatial dependence on traveling-wave amplification (Fig. S2A,D). As we demonstrate in the model, this conclusion holds even when the local micromechanical processes that regulate the activity of the cochlear amplifier are perfectly scaling symmetric. Furthermore, the result does not depend on the particular model form of the CP admittance-all active cochlear models necessarily include processes that modify both the real and imaginary parts of the wavenumber (see [16]). Figure S5 illustrates   of the tapered cochlea, with those obtained in the corresponding passive model (obtained by setting τ = 0 in Eq. S.23). Interestingly, the model predicts that the active process causes the in vivo near-CF gain relative to that measured post mortem, to decrease with distance from the stapes. These model results, which are consistent with both classic and more recent experimental recordings from the apex [12,10], are solely due to the hydrodynamics of the tapered cochlea. Interestingly, at apical locations, the model also predicts the existence of "active gain" well above CF, a phenomenon recently observed in recordings from the apex of the guinea-pig cochlea [10].